FACTORIALS AND THE RAMANUJAN FUNCTION

In 2006, F. Luca and I. E. Shparlinski (Proc. Indian Acad. Sci. (Math. Sci.) 116(1) (2006), 1-8) proved that there are only finitelymany pairs (n, m) of positive integers which satisfy the Diophantine equation vertical bar tau (n!)vertical bar = m!, where tau is the Ramanujan function. In this paper, we follow the same approach of Luca and Shparlinski (Proc. Indian Acad. Sci. (Math. Sci.) 116(1) (2006), 1-8) to determine all solutions of the above equation. The proof of our main theorem uses linear forms in two logarithms and arithmetic properties of the Ramanujan function.